What is XOR (Exclusive-OR (Logical Operation))
XOR: Exclusive OR (Logical Operation)
XOR (Exclusive OR) is a logical operation that outputs true (1) when an odd number of inputs are true, and false (0) otherwise. It's denoted by the symbol ⊕.
Truth Table
| Input A | Input B | Output (A XOR B) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
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Properties of XOR
- Commutative: A ⊕ B = B ⊕ A
- Associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)
- Identity element: A ⊕ 0 = A
- Inverse element: A ⊕ A = 0
- Linearity: A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C = A ⊕ B ⊕ C
XOR Gate
An XOR gate is a digital logic gate that performs the XOR operation. It has two inputs and one output.
Opens in a new windowanalog.comXOR gate
Applications of XOR
- Error Detection: XOR is used in parity checking to detect single-bit errors in data transmission.
- Cryptography: XOR is used in various cryptographic algorithms, such as encryption and decryption.
- Hamming Code: XOR is used in Hamming codes for error correction.
- Checksum Calculation: XOR is used in checksum algorithms to verify data integrity.
- Arithmetic Operations: XOR is used in binary addition without carry.
- Pseudo-Random Number Generation: XOR can be used in linear feedback shift registers (LFSRs) for generating pseudorandom sequences.
Example: Parity Checking
Let's consider a simple example of parity checking using XOR. Suppose we have a data block: 10110. To calculate the parity bit, we perform XOR on all bits:
1 XOR 0 XOR 1 XOR 1 XOR 0 = 1
The parity bit is 1. If a single bit error occurs during transmission, the calculated parity bit at the receiver will not match the transmitted parity bit, indicating an error.
Conclusion
XOR is a fundamental logical operation with wide-ranging applications in computer science, digital electronics, and cryptography. Its ability to detect differences and its properties make it a valuable tool in various algorithms and systems.